PQ is the diameter of a circle with centre A. M is a point on the circumference of the circle. If the measure of angle MAP is 110°, then find the measure of angle MQA.

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Given: PQ is the diameter of the circle with center A, and M is a point on the circumference. We have ∠MAP = 110°.

Step 1: Since A is the center and PQ is the diameter, points P and Q lie on the circle such that PQ passes through A.

Step 2: ∠MAP is the angle at the center formed by points M and P. The angle subtended by the chord MP at the center is 110°.

Step 3: The angle subtended by the same chord MP at the circumference (on the opposite side) is half the angle at the center. This is a property of circles: the angle subtended by a chord at the center is twice the angle subtended at the circumference.

Step 4: Therefore, ∠MQA = 1/2 × ∠MAP = 1/2 × 110° = 55°.

Hence, the measure of angle MQA is 55°.

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